Figure  REFA shows a micrograph of device B with X \( \approx 240 \, {nm}\) , for which a QD is formed in the tunnel junction as sketched in Fig.  REFB. In this QD-SC system, two ground states (GSs) are accessible: a spin doublet, \( \mathinner {|{D}\rangle }\) , with spin \(\frac{1}{2}\) , and a spin singlet, \( \mathinner {|{S}\rangle }\) , with spin 0.
Whether the GS is a doublet or a singlet is determined by the interplay between the Andreev processes at the SC-QD interface and Coulomb blockade (CB), with charging energy \(U\) , in the QD. CB enforces a one by one electron filling, favouring odd electron occupations with a doublet GS [1]}, [2]}, [3]}, [4]}. The coupling to the SC, \(\Gamma _S\) , on the other hand, privileges a singlet GS. Its physical nature crucially depends on the ratio \(\Delta /U\) . In the large \(\Delta /U\)  limit, the coupling to the SC mainly induces local superconducting correlations in the QD that lead to Bogoliubov-type singlets, which are superpositions of the empty \(\mathinner {|{0}\rangle }\)  and doubly-occupied \(\mathinner {|{\uparrow \downarrow }\rangle }\)  states in the QD. In the opposite, small \(\Delta /U\)  limit, the unpaired spin in the QD couples with quasiparticles in the SC, see Fig.  REFB, with an exchange interaction \(J\sim 2\Gamma _S/U\) . This exchange interaction creates so-called Yu-Shiba-Rusinov (YSR) singlets, the superconducting counterpart of Kondo singlets [5]}, [6]}, [7]}, [8]}. For small \(J\) , the GS is a doublet and the YSR singlets occur as discrete ABS excitations near the edge of the SC gap. A larger \(J\) , however, detaches these excitations at subgap energies, and eventually may induce zero-energy crossings, which signal a quantum phase transition (QPT) where the YSR singlet becomes the new GS. Since our experiments are always in a large \(U\)  limit (even for the largest gap at \(\phi =0\)  we always have \(U\gg \Delta \) ), the YSR regime [9]}, [10]}, [11]}, [12]}, [13]}, [4]}, [15]} is the relevant one to explain our data (for a full theoretical discussion about all the physical regimes, see the SI). Transitions between the GS and the first excited state of the system, i.e., between a doublet and a singlet state or vice-versa, manifest in transport spectroscopy as a subgap resonance at voltage \(V=\zeta \)  (and its electron-hole-symmetric partner at \(V=-\zeta \) ). Furthermore, changes in the parity of the GS of the system appear as points in parameter space (here \(V_{\textrm {bg}}\)  or \(B\) ) where \(\zeta \)  changes sign (signalled by the crossing of ABSs at zero energy)  [9]}, [10]}, [11]}, [12]}.
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